Big Numbers

The other day, in the graduate student talks, Subhojoy was talking about the Simultaneous Uniformization Theorem. It was a nice treat because I used to be really into geometric topology (or at least as much as an undergrad with way too little background could.)

The big reveal is
but most of the talk, naturally, goes into defining what those letters mean.

The Riemann Mapping Theorem says that any simply connected set is conformally equivalent to the disc . Conformal maps are angle-preserving; in practice, they’re holomorphic functions with derivative everywhere nonzero. Conformal maps take round circles to round circles if the circles are small enough.

A Riemann Surface is a topological surface with conformal structure: a collection of charts to the complex plane such that the transition maps are conformal.

The first version of the Uniformization Theorem says that any simply-connected Riemann surface is conformally equivalent to the Riemann sphere (or complex plane or unit disc; these are all equivalent.)

The second, more general version of the Uniformization Theorem says that any Riemann surface of genus is conformally equivalent to where is the hyperbolic plane and is a discrete subgroup of .

To understand this better, we should observe more about the universal cover of a Riemann surface. This is, of course, simply connected. Its deck transformations are conformal automorphisms of the disc. But it can be proven that conformal automorphisms of the disc are precisely the Mobius transformations, or functions of the form

This implies that the automorphism group of is .

Now observe that there’s a model of the hyperbolic plane on the disc, by assigning the metric

And, if you were to check, it would turn out that conformal transformations on the disc preserve this metric.

So it begins to make sense; Riemann surfaces are conformally equivalent to their universal covering space, modulo some group of relations, a subgroup of the group of deck transformations of the universal cover.

are called Fuchsian groups — these define which Riemann surface we’re talking about, up to a conformal transformation.

Now we can define Fuchsian space as

It’s the set of maps from the fundamental group of a surface to .

And we can define Teichmuller space as the space of marked conformal structures on the surface S.
This is less enormously huge than you might think, because we consider these up to an equivalence relation. If and are conformal structures, and there exists a conformal map such that then we consider equivalent structures.

In fact, Teichmuller space is not that enormously huge: . It turns out that Teichmuller space is completely determined by what happens to the boundary circles in a pair of pants decomposition of the surface.

Here’s a picture of a pair of pants (aka a three-punctured sphere):

Here’s a picture of a decomposition of a Riemann surface into pairs of pants:

(Here’s a nice article demonstrating the fact. It’s actually not as hard as it looks.)

Now we generalize to Quasi-Fuchsian spaces. For this, we’ll be working with hyperbolic 3-space instead of 2-space. The isometries of hyperbolic 3-space happen to be .
Instead of a Poincare Disc Model, we have a ball model; again, acts by Mobius transformations, functions of the form .

A quasiconformal function takes, infinitesimally, circles to ellipses. It’s like a conformal map, but with a certain amount of distortion. The Beltrami coefficient definds how much distortion:

Quasi-Fuchsian space, QF(S), is the set of all quasiconformal deformations of a Fuchsian representation. In other words, this is the space of all representations to preserving topological circles on the boundary.

Now, the Simultaneous Uniformization Theorem can be stated:
the Quasi-Fuchsian space of a surface is isomorphic to the product of two Teichmuller spaces of the surface.

One application of this theorem is to hyperbolic 3-manifolds.
If is a hyperbolic 3-manifold, and if then $M \simeq S \times \mathbb{R}$.

In other words, we can think of a hyperbolic three-manifold as the real line, with a Riemann surface at each point — you can only sort of visualize this, as it’s not embeddable in 3-space.

The Simultaneous Uniformization Theorem implies that there is a hyperbolic metric on this 3-manifold for any choice of conformal structure at infinity.

This contrasts with the Mostow Rigidity Theorem, which states that a closed 3-manifold has at most one hyperbolic structure.

Together, these statements imply that any hyperbolic metric on is determined uniquely by the choice of conformal structures at infinity.

This is a method for using topological invariants to define data. In particular, we want to define the homology of a data set representing a point cloud in a high-dimensional Euclidean space. A topologist would replace the point cloud with a simplicial complex — one of the easiest to compute is the Rips Complex, whose k-simplices correspond to unordered k+1-tuples of points within pairwise Euclidean distance .

However, the resulting simplicial complex depends on the choice of . A very small leaves the complex a discrete set, while a very large results in the complex being one big simplex. As moves, holes in the simplicial complex are born, grow, and die; the picture over time provides a description of the data’s topology.

The “persistence complex” is a sequence of chain complexes together with chain maps , which are inclusions. This is motivated by an increasing sequence of s and the inclusions from one complex to the next. (The precision here goes from fine to coarse.)

Ghrist introduces the notion of a “barcode” — each “bar” is a generator of the homology group, and the length of the bar is the range of values of for which this particular element is a generator of the homology group. A barcode is the persistence analogue of a Betti number.

Now, what I always wondered here is what this has to do with geometry. Consider a finger-shaped projection sticking out of a 2-dimensional surface. At different resolutions, the projection can appear to break off into an island. (This is a practical problem for protein visualization.) This would be an example of a feature that could be captured with persistence homology; but it could also be explained directly by noticing that the tip of the projection is a region of high curvature. Could other persistence homology features be explained by geometrical properties?

This paper by Gunnar Carlsson et al, seems to provide an answer. The authors define a tangent complex of a space, which is the closure of the set of all tangents to points in the space. Then, they define the filtered tangent complex, which is the set of tangent vectors for which the osculating circle has a radius larger than some . We have an inclusion between filtered tangent complexes of different s. (For curves, there is only one osculating circle; for surfaces, there is one in each direction, so the tangent space is defined based on the maximum between them.)

Then we look at the homology groups of the filtered tangent spaces. This provides a barcode. Such barcodes can, for instance, distinguish a bottle from a glass. (The relationship of the tangent-space barcode to the Rips-complex barcode remains mysterious to me.)